Twin Prime Conjecture

There are two related conjectures, each called the twin prime conjecture. The first version states that there are an infinite number of pairs of twin primes (Guy 1994, p. 19). It is not known if there are an infinite number of such primes (Wells 1986, p. 41; Shanks 1993, p. 30), but it seems almost certain to be true. While Hardy and Wright (1979, p. 5) note that "the evidence, when examined in detail, appears to justify the conjecture," and Shanks (1993, p. 219) states even more strongly, "the evidence is overwhelming," Hardy and Wright also note that the proof or disproof of conjectures of this type "is at present beyond the resources of mathematics."

Arenstorf (2004) published a purported proof of the conjecture (Weisstein 2004). Unfortunately, a serious error was found in the proof. As a result, the paper was retracted and the twin prime conjecture remains fully open.

The conjecture that there are infinitely many Sophie Germain primes, i.e., primes such that is also prime, is very closely related (Shanks 1993, p. 30).

A second twin prime conjecture states that adding a correction proportional to to a computation of Brun's constant ending with will give an estimate with error less than . An extended form of this conjecture, sometimes called the strong twin prime conjecture (Shanks 1993, p. 30) or first Hardy-Littlewood conjecture, states that the number of twin primes less than or equal to is asymptotically equal to

where is the so-called twin primes constant (Hardy and Littlewood 1923). The value of is plotted above for , with indicated in blue and taking .

This conjecture is a special case of the more general k-tuple conjecture (also known as the first Hardy-Littlewood conjecture), which corresponds to the set .

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